Methods and apparatus for rendering or preparing digital objects or portions thereof for subsequent processing

ABSTRACT

Methods and apparatus render images of digital objects or prepare digital objects for subsequent processing. The method includes sorting data representative of positions of at least three vertices of polygons of a digital object, then determining whether the orientation of the vertices of each polygon from a specific reference point differs from the actual, or original, orientation of the vertices. Such a determination may be made by generating an orientation decision variable based on the relative positions of the vertices and calculating a cross product term (CPT) after the vertex data has been sorted. The CPT may also be used in other operations involving the polygon, such as in imparting appearance characteristics to the polygon. The method may be embodied as a computer program that controls the operation of a processor. Accordingly, processors, computers, and systems that render images of digital objects in accordance with the method are also disclosed.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of application Ser. No. 11/210,402,filed Aug. 24, 2005, now U.S. Pat. No. 7,619,624 issued Nov. 17, 2009,which is a continuation of application Ser. No. 09/888,890, filed Jun.25, 2001, now U.S. Pat. No. 6,937,236, issued Aug. 30, 2005. Thedisclosure of each of the foregoing documents is incorporated herein inits entirety by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to methods and apparatus forrendering digital objects and, more specifically, to methods foridentifying polygons that are back facing relative to a reference pointand for culling the data of such back facing polygons prior torendering. In particular, the methods and apparatus of the presentinvention decide whether to cull data for a particular polygon after thevertices of that polygon have been sorted to facilitate the acceleratedprocessing of the polygons.

2. Background of the Related Art

As is well known in the art, computers typically break three-dimensionalobjects down into a series of polygons (e.g., triangles), which togethermake up the outer surfaces of the three-dimensional object. The relativepositions of each of the polygons of a three-dimensional object aretypically stored and manipulated in digital form, i.e., a series of 0'sand 1's that indicates the locations of the polygons relative to oneanother on the three-dimensional object, as well as the positions, orlocations (i.e., x-, y-, and z-coordinates), of the vertices of eachpolygon of the three-dimensional object. Accordingly, for purposes ofthis disclosure, the digital form of a three-dimensional object isreferred to as a “digital object.”

The data for each polygon of a digital object typically includespositional data for at least three vertices of the polygon along twoadjacent edges. If the polygon is a triangle, the digital make-up of thetriangle includes positional data for all three vertices of thetriangle. Typically, the positional data for the vertices of eachpolygon of the digital object are arranged in the same orientation, ororder. For example, the positional data for the vertices of all of thepolygons of a digital object may be ordered around the peripheries ofthe polygons in a clockwise (CW) orientation or in a counterclockwise(CCW) orientation.

In addition to breaking a three-dimensional object down to datarepresentative of the vertices of a collection of polygons havingspecific relative positions, characteristic data (e.g., color, shading,texture) for the entire three-dimensional object, for a portion thereof,or for each polygon or a small set of polygons may also be stored andprocessed in digital form.

When viewed from a specific reference point, such as from the viewpointof a computer user looking at a video monitor or from one looking at animage on paper, a digital three-dimensional object is shown in twodimensions. Typically, only the portions of the digital object thatface, or that are visible from, the reference point are shown. Theseportions of the digital object are the so-called “front facing” polygonsof the digital object. This is true both when the digital object is in astationary position and when the digital object is being moved ormanipulated.

Polygons of a digital object that do not face the reference point aretypically referred to as “back facing” polygons. Relative to thereference point, the vertices of a back facing polygon have anorientation that is opposite the actual orientation of the vertices ofthat polygon. For example, if a polygon with CW-oriented vertices isback facing, the vertices will appear, from the reference point, to havea CCW orientation.

As back facing polygons of a digital object do not face a referencepoint, such as a viewer, back facing polygons are typically notdisplayed to the viewer. Accordingly, prior to rendering eachsynthesized, or rendered, image (i.e., each particular orientation orstate), or scene, of the digital object, the other, nonvisible, backfacing polygons of the digital object are typically culled, oreliminated from further processing until an image of the digital objecthas been rendered. By culling data for back facing polygons, renderingof an image of the digital object will be accelerated.

Prior to rendering an image of the digital object, the positional datafor the vertices of a polygon are typically evaluated to determinewhether, relative to the reference point of a viewer, the polygon isfront facing or back facing.

Once the polygons of an image or scene of a digital object aretransformed into the viewer's perspective (i.e., a two-dimensionalimage), the sign (S) of the z-axis term of the cross product, or crossproduct term (CPT), of two edges of each transformed polygon willindicate the orientation of vertices of the polygon (e.g., whether thetransformed polygon has a CW or CCW orientation). For example, if thevertices of a polygon are oriented CW, the actual, or original, sign bit(S₀) is a “1.” If the vertices of a polygon are oriented CCW, theactual, or original, sign bit (S₀) is a “0.” This sign bit is then usedto compare the (two-dimensional) position of each polygon in the imageto be rendered to its actual, or original, orientation on the object(its two-dimensional orientation when front facing). If theseorientations are not equal, the polygon is determined to be a backfacing polygon and the data is culled from the rendering process.

An example of a conventional method for back face culling includesselecting the vertices, V0, V1, and V2, along two adjoined edges of eachpolygon. For purposes of this disclosure, V0, V1, and V2 refer to theorder in which the positional data for the vertices of a polygon areintroduced into a rendering application; these designations do notnecessarily refer to the order in which vertices are located around theperiphery of the polygon. The two-dimensional x-axis and y-axispositions of each of the vertices of a polygon of a digital object to bedisplayed or otherwise output are then determined, with V0.x, V1.x, andV2.x being the respective x-axis positions of vertices V0, V1, and V2and V0.y, V1.y, and V2.y being the respective y-axis positions ofvertices V0, V1, and V2. The differences between the x-axis and y-axispositions of each of the selected vertices are then determined, forexample, as follows:X10=V1.x−V0.x;Y12=V1.y−V2.y;X12=V1.x−V2.x; andY10=V1.y−V0.y.The CPT for the polygon is then calculated, as follows:CPT=(X10×Y12)−(X12×Y10).Assuming that the digital order in which the positional data for theanalyzed vertices, V0, V1, and V2, are arranged remains unchanged fromthe order in which the positional data for these vertices was introducedinto the rendering application, the sign (S) of the CPT (S=sign(CPT)) isthen used to determine whether or not the data for the analyzed polygonshould be culled prior to rendering. If the actual orientation of thevertices of the polygon was CW (i.e., S₀=1) and S=0, the analyzedpolygon is back facing and, accordingly, the positional data for thatpolygon is culled. Alternatively, if the actual orientation of verticesof the polygon was CCW (i.e., S₀=0) and S=0, the positional data for thevertices of the polygon is retained for rendering purposes, as thepolygon is front facing relative to the reference point and the polygonwill be shown when the digital object is displayed or otherwise output.

In rendering a two-dimensional image of a three-dimensional digitalobject, it is often necessary to sort the vertices of each polygon. Forexample, the positional data for the vertices of a particular polygon ofthe digital object may be introduced into the rendering application in aparticular order, which is then changed, such as in a triangle setupoperation, to facilitate rendering of the polygon. Accordingly, thepositional data for the various vertices of the polygon may bere-ordered, or shuffled, which will change a CPT that is based on theorder in which the positional data for the vertices is presented. Thus,while conventional methods for culling back facing polygons decrease theamount of time necessary to render an image of a digital object, thevertex differences or CPTs calculated by these methods cannot bereliably reused for subsequent rendering operations, such as polygoncharacterization processes (e.g., determining the coloration, shadowing,or texturing of a polygon). As a consequence, the vertex differences orCPTs must be recalculated once sorting has occurred.

Moreover, if the positional data for the vertices of a particularpolygon are sorted prior to determining the vertex differences or CPTs,without having the positional data for each vertex tied to an indicatorof the position of the vertex on the polygon, the sign of a subsequentlycalculated CPT could provide incorrect information about whether thepolygon is front facing or back facing.

No known prior art method or apparatus uses a CPT obtained by evaluatingsorted vertices that can be used in analyzing polygons of a graphicimage and determining whether to cull data representative of eachanalyzed polygon. Similarly, no known method or apparatus is used forcalculating a CPT that may be used both in decisions to cull back facingpolygons of a digital object and in subsequent rendering operations.

SUMMARY OF THE INVENTION

The present invention includes a method for using sorted vertices tomake a decision on whether to cull back facing polygons. Apparatus andsystems using the method to display one or more images of a digitalobject are also within the scope of the present invention.

The method of the present invention includes sorting the vertices of atleast one polygon of a digital object, determining a CPT for the sortedvertices of the polygon, and determining whether or not the sign of theCPT should be reversed, or multiplied by −1.

Methods incorporating teachings of the present invention facilitate theuse of sorted vertices in calculating the positional differences betweenthe sorted vertices, as well as the use of sorted vertices incalculating a CPT for the sorted vertices. The sign of the CPT may thenbe used to determine whether the analyzed polygon is back facing and, ifso, to indicate a decision to cull data representative of the polygonfrom the process of rendering an image of a digital object of which theback facing polygon is a part. In addition, the positional differencesand CPT calculated in accordance with teachings of the present inventionfor each analyzed polygon may also be used in other operations relatingto the processing or rendering of that polygon and of a digital image ofwhich that polygon is a part.

The data for the vertices of each polygon of a digital object may besorted by known processes, such as those employed in triangle setupprocedures. Once the data is sorted, a CPT for the data may bedetermined, also using known processes.

Sorted and potentially rearranged vertices are used in accordance withthe method of the present invention to determine the positionaldifferences between the selected vertices of each analyzed polygon of adigital object, as well as the CPT of the selected vertices.Nonetheless, because the order of the selected vertices of each analyzedpolygon may have been rearranged, the positional differences betweenadjacent ones of the selected vertices of each analyzed polygon and theCPT of each analyzed polygon may not indicate whether an analyzedpolygon is a front facing or back facing polygon. Rather, the sign ofthe CPT may indicate that a front facing polygon is back facing or thata back facing polygon is front facing. The method of the presentinvention, therefore, includes a process wherein the order into whichthe data for the vertices of the polygon has been sorted is analyzed todetermine whether or not the sign of the CPT should be changed, orinverted, prior to making a back face culling decision. An exemplaryembodiment of this process includes computing a decision variable, T, todetermine whether to change the sign of the CPT (e.g., from positive tonegative or from 1 to 0, or from negative to positive or from 0 to 1).

Since the CPT and the values of the positional differences between thevertices of each analyzed polygon are based on sorted vertex data, thesevalues may be used in other operations, such as coloring, shading, andtexturing, that are conducted on each front facing polygon prior torendering an image of a digital object.

Other features and advantages of the present invention will becomeapparent to those of ordinary skill in the art through consideration ofthe ensuing description, the accompanying drawings, and the appendedclaims.

BRIEF DESCRIPTION OF THE DRAWINGS

In the drawings, which illustrate exemplary embodiments of the presentinvention:

FIG. 1 is a schematic illustration of a three-dimensional digitalobject, including, from the reference point of a viewer's perspective,front facing polygons, shown in solid lines, and back facing polygons,which are not shown;

FIG. 2 is a flow diagram that illustrates the method of the presentinvention;

FIG. 3 schematically depicts a polygon of the digital object illustratedin FIG. 1; and

FIG. 4 is a schematic representation of a system and apparatus thatoperate under control of a program incorporating methods of the presentinvention.

DETAILED DESCRIPTION OF THE INVENTION

With reference to drawing FIG. 1, a three-dimensional digital object 10is schematically illustrated. At least an outer surface 12 of digitalobject 10 is made up of a plurality of polygons 14, which areillustrated as being triangles having substantially identicaldimensions. Each polygon 14 of digital object 10 includes at least threevertices 16, or corners. The numeric data stored for each polygon 14 ofdigital object 10 represents the positions of each vertex 16 of polygon14 in the x-, y-, and z-axes. Data representative of the positions ofthe at least three vertices 16 of all of the polygons 14 that make up atleast outer surface 12 of digital object 10 are used in variousprocesses, including movement or manipulation of digital object 10, aswell as in rendering an image of digital object 10.

A digital object 10 is rendered to display an image of digital object 10to a viewer, such as on a computer monitor or on a printed page. Oncethe digital object 10 has been manipulated or moved relative to areference point, such as the perspective of a viewer, the digital object10 must be rerendered to indicate movement thereof to the viewer.

In rendering an image of a digital object 10, data regarding theorientation of digital object 10 and of the polygons 14 thereof isevaluated and data representative of the positions of the vertices 16 ofeach polygon 14 of digital object 10 is used to “build” an image ofdigital object 10 as it is to be shown to a viewer, such as on acomputer monitor or on a printed page. The present invention includesmethodology for effecting a portion of the rendering process, whereinthe vertex 16 data for each polygon 14 of an image of digital object 10is sorted, then the data for vertices 16 of polygons 14 that will not bevisible from the reference point of a viewer (e.g., a computer user)will be culled, or omitted, from the rendering process, thus consumingless random access memory of a computer and speeding up the renderingprocess.

Illustrated in drawing FIG. 2 is a flow diagram illustrating the sortingand back face cull decision-making processes of the present invention.

Illustrated in drawing FIG. 3 is a graphic representation of an analyzedpolygon 14 of a digital object 10 (see FIG. 1). As illustrated, polygon14 is a triangle and includes three vertices 16 a, 16 b, and 16 c.

With reference to both drawing FIGS. 2 and 3, prior to rendering animage of digital object 10, data for each vertex 16 of each polygon 14of digital object 10 are sent to a portion of a processor under controlof a sorting application, as shown at reference character 110 of FIG. 2.The order in which data for each of vertices 16 a, 16 b, 16 c of polygon14 are sent to the processor to be sorted may or may not be arbitrary.For example, the positional data for each of the vertices of a polygonmay be organized in an order that facilitates a determination of newpositions of each vertex following movement of the digital object. Whilethe positional data for each of the three or more vertices of a polygonis “tied to,” or associated with, that polygon, in order to conservememory, the positional data may not be associated with an indicator fora position of a vertex on the polygon relative to other vertices of thepolygon; rather, corresponding x-axis, y-axis, and z-axis data could beassociated with a particular vertex based merely on the relative bitlocations of these data.

Data representative of the actual CW or CCW orientation of vertices 16a, 16 b, 16 c of all of the polygons 14 of digital object 10 may also besent to the processor, also at reference character 110 of drawing FIG.2.

As the data for vertices 16 a, 16 b, 16 c may be sent to the processorin an arbitrary order, the data representative of the positions ofvertices 16 a, 16 b, 16 c are then sorted, or organized or ordered, asindicated at reference character 112 of drawing FIG. 2, in a manner thatfacilitates further processing and use of data representative of thepositions of vertices 16 a, 16 b, 16 c and other values based thereon inrendering of an image of digital object 10.

For example, in triangle setup, the coordinates of vertices 16 a, 16 b,16 c are rasterized so the locations of the lines that define edges 18a, 18 b, 18 c of each polygon 14 of digital object 10 may be determinedand front facing polygons 14 may be rendered. Rasterization couldproceed in any order (e.g., horizontally from top to bottom or frombottom to top, or vertically from left to right or from right to left).Accordingly, the order of the vertices 16 a, 16 b, 16 c is determinedbased on the order of rasterization. To compute the lines, or polygonedges 18 a, 18 b, 18 c, between vertices 16 a, 16 b, 16 c of polygon 14,the relative positions of vertices 16 a, 16 b, 16 c must be determinedand the order in which positional data representative of each vertex 16is processed may be changed, depending on the order of rasterization. Asan example of the manner in which vertices 16 may be organized, thefollowing series of equations may be used:

If ((V2.y == V0.y) AND (V2.x < V0.x)) SORT[2] = 1 Else if (V2.y < V0.y)SORT[2] = 1 Else SORT[2] = 0 If ((V1.y == V2.y) AND (V1.x < V2.x))SORT[1] = 1 Else if (V1.y < V2.y) SORT[1] = 1 Else SORT[1] = 0 If ((V1.y== V0.y) AND (V1.x < V0.x)) SORT[0] = 1 Else if (V1.y < V0.y) SORT[0] =1 Else SORT[0] = 0This series of equations arranges the positional data for vertices 16 ofpolygon 14 in order of their relative vertical positions. If twovertices 16 have the same vertical orientation, these two vertices 16are then organized according to the relative horizontal positions.

As the CPT for a particular polygon 14 of a digital object 10 or thesign of the CPT may depend on the order in which the positional data foreach of vertices 16 a, 16 b, 16 c of polygon 14 is introduced into aprocessor, sorting of the vertex data may change one or both of the CPTand the sign thereof. As a consequence, a CPT that is calculated beforepositional data for vertices 16 of polygon 14 is sorted may not beuseful in processes that follow a back face culling decision.

Once sorting has occurred, or concurrently with sorting the vertices 16of an analyzed polygon 14 of a digital object, as depicted at referencecharacter 114 of drawing FIG. 2, an orientation decision variable may becomputed (e.g., by the processor) to provide information about whetheror not the sign of a CPT for an analyzed polygon 14, which has yet to becalculated until after the positional data for vertices 16 of analyzedpolygon 14 has been sorted, will accurately indicate the orientation ofvertices 16 of analyzed polygon 14. If the orientation decision variableindicates that the sign of the CPT will not accurately indicate theorientation of vertices 16 of analyzed polygon 14, the orientationdecision variable for polygon 14, which is to be subsequently used in aback face culling decision to denote that the apparent orientation ofvertices 16 of analyzed polygon 14, as indicated by the sign of the CPT,is opposite the actual orientation of vertices 16 of polygon 14 whenviewed from a particular reference point R. Accordingly, the orientationdecision variable indicates that the actual orientation of vertices 16of polygon 14 is inverted from the apparent orientation of vertices 16,as indicated by the sign of the CPT.

As evidenced in the exemplary set of equations provided above, theorientation decision variable may be a three-bit variable that isdetermined substantially concurrently with sorting of the positionaldata for each vertex 16 of a polygon 14. By way of example, thefollowing table illustrates, for each sort indication variable, theorder in which vertices 16 of polygon 14 have been sorted (from sortstep 2 to sort step 0, or [2:0]), relative to the presortingorientation, as well as the corresponding one bit orientation decisionvariable (T):

SORT [2:0] Relative Vertex Order T 000 V0, V2, V1 1 001 N/A N/A 010 V0,V1, V2 0 011 V1, V0, V2 1 100 V2, V0, V1 0 101 V2, V0, V1 1 110 N/A N/A111 V1, V2, V0 0

Based on the sorted vertices, a CPT may be calculated (e.g., by theprocessor), as indicated at reference character 116 of drawing FIG. 2.The sign of the CPT, which is determined (e.g., by the processor), asknown in the art, at reference character 118 of drawing FIG. 2, alongwith the orientation decision variable, may then be used in a back faceculling decision (e.g., by the processor), as depicted at referencecharacter 120 of drawing FIG. 2, to determine whether the orientation ofan analyzed polygon 14 has changed from its actual orientation. Forexample, if vertices 16 on each polygon 14 of a digital object 10 areactually oriented CW, the sign of the CPT indicates a CCW orientation,and the decision variable (e.g., a “1”) indicates that sorting hasswitched the sign of the CPT, the data for an analyzed polygon 14 willbe used in rendering an image with that polygon 14 displayed.Alternatively, if, for a polygon 14 that is actually CW-oriented, thesign of the CPT indicates a CCW orientation and the decision variable(e.g., a “0”) indicates that sorting has not altered the sign of theCPT, the sign of the CPT and the orientation decision variable togetherindicate that an analyzed polygon 14 is a back facing polygon and thedata for polygon 14 is, therefore, culled.

A CPT that has been calculated following sorting of the data may also beused in processing that follows a sorting operation such as trianglesetup but that must occur before the digital object is rendered. Forexample, but not to limit the scope of the present invention, a CPT thatis calculated after sorting of the positional data for the vertices of apolygon may be used, as known in the art, to process variouscharacteristic parameters of the polygon, such as the shadowing, color,or texture of the polygon, based on the position of the polygon on thedigital object relative to a specific reference point R.

Accordingly, methods incorporating teachings of the present inventionfacilitate the faster processing of polygon data, as well as a reductionin the amount of memory required to process the polygon data.

As the method of the present invention is particularly useful whenembodied as a set of instructions to control various decisions made bylogic circuits of a computer processor, the present invention alsoincludes apparatus and systems that process and display one or moreimages of digital object 10 or a portion thereof in accordance withteachings of the present invention. Illustrated in drawing FIG. 4 is anexemplary embodiment of such a system 200 and its component apparatus.For example, a system 200 incorporating teachings of the presentinvention may include a computer 210 with a processor 212 and memory 214associated therewith, as well as hardware and programming configured tooutput one or more full or partial images of a digital object 10 to anapparatus that facilitates viewing of the image by a user, such as amonitor 216 or a printer 218.

Various logic circuits of processor 212 may be configured or programmed,such as by a back face culling application, to perform certain tasks inaccordance with the inventive method. For example, a first logic circuitof processor 212 may sort data representative of at least three vertices16 (FIGS. 1 and 3) of at least one polygon 14 (FIGS. 1 and 3) of digitalobject 10 (FIG. 1), a second logic circuit of processor 212 may beconfigured to generate an orientation decision variable based onrelative positions of vertices 16, and a third logic circuit ofprocessor 212 may calculate a CPT of vertices 16 following sortingthereof by the first logic circuit. The first and second logic circuitsof processor 212 may operate substantially concurrently, or at differenttimes. When under control of a program that effects the method of thepresent invention, processor 212 may also include a fourth logic circuitthat determines a sign of the CPT. Processor 212 may also include afifth logic circuit that determines an orientation of one or morepolygons 14 of digital object 10 based on the sign of the CPT and theorientation decision variable. The fifth logic circuit may also decidewhether to cull data of vertices 16 based on the orientation of thecorresponding polygon 14.

These logic circuits may comprise completely separate circuits ofprocessor 212, or any combination of these logic circuits may bepartially combined.

Although the foregoing description contains many specifics, these shouldnot be construed as limiting the scope of the present invention, butmerely as providing illustrations of some of the presently preferredembodiments. Similarly, other embodiments of the invention may bedevised which do not depart from the spirit or scope of the presentinvention. Moreover, features from different embodiments of theinvention may be employed in combination. The scope of the invention is,therefore, indicated and limited only by the appended claims and theirlegal equivalents, rather than by the foregoing description. Alladditions, deletions, and modifications to the invention, as disclosedherein, which fall within the meaning and scope of the claims are to beembraced thereby.

1. A method, comprising: storing data representative of polygons of adigital object in a memory of a computing system; sorting at least threevertices of at least one polygon based on relative positions of the atleast three vertices along at least one axis; calculating a crossproduct term of the at least one polygon after the sorting; generatingan orientation decision variable based on the relative positions of theat least three vertices; evaluating a sign of the cross product termrelative to the orientation decision variable; and culling at least someof the data representative of the polygons from the memory responsive tothe evaluating.
 2. The method of claim 1, wherein the sorting at leastthree vertices of at least one polygon further comprises sorting the atleast three vertices along at least one additional axis.
 3. The methodof claim 1, wherein the evaluating comprises determining a back facingorientation or a front facing orientation of the at least one polygon.4. The method of claim 1, further comprising changing the sign of thecross product term if the sign does not correspond to the orientation ofthe at least one polygon.
 5. The method of claim 1, wherein the sortingand the generating are effected concurrently.
 6. A method, comprising:sorting data representative of one or more polygons of a digital objectbased on relative positions of at least three vertices of the one ormore polygons along at least one axis; calculating a cross product termof the one or more polygons after sorting the data; generating anorientation decision variable for the one or more polygons based on arelative order of the at least three vertices; evaluating a sign of thecross product term for the one or more polygons relative to theorientation decision variable; and rendering an image of at least aportion of the digital object on an apparatus for viewing the digitalobject.
 7. The method of claim 6, wherein the sorting the data is basedon relative vertical positions of the at least three vertices.
 8. Themethod of claim 6, wherein the sorting the data further comprisessorting along at least one additional axis orthogonal to the at leastone axis.
 9. The method of claim 6, wherein the calculating includesdetermining positional differences between adjacent vertices of the oneor more polygons after the sorting the data.
 10. The method of claim 6,further comprising using at least one of the cross product term andvalues of positional differences between the at least three vertices inat least one of a coloring process, a shading process, and a texturingprocess for the one or more polygons.
 11. A system, comprising: a memoryfor storing computing instructions; and a processor operably coupled tothe memory and configured to execute the computing instructions for eachpolygon of a plurality if polygons representing a digital object, to:sort at least three vertices of the polygon based on relative positionsof the at least three vertices along at least one axis; calculate across product term of the polygon after the sorting; generate anorientation decision variable based on the relative positions of the atleast three vertices; and determine if the polygon is back facingresponsive to a comparison of the orientation decision variable to asign of the cross product term.
 12. The system of claim 11, furthercomprising a display apparatus operably coupled to the processor andwherein the processor is further configured to execute the computinginstructions to: cull from the digital object each polygon of theplurality of polygons that is back facing; and after the culling, renderan image of at least a portion of the digital object on the displayapparatus.
 13. The system of claim 11, wherein the processor is furtherconfigured to execute the computing instructions to further process eachpolygon of the plurality of polygons with one or more of shadowing,coloring, and texturing.
 14. An electronic system, comprising: means forsorting vertex data representative of at least one polygon of a digitalobject based on relative positions of at least three vertices along atleast one axis; means for generating an orientation decision variablebased on the relative positions of the at least three vertices; meansfor determining a sign of a cross product term of the at least onepolygon after operation of the means for sorting; and means forcomparing the sign of the cross product term and the orientationdecision variable.
 15. The electronic system of claim 14, wherein themeans for comparing comprises means for determining a back facingorientation or a front facing orientation of the at least one polygon.16. The electronic system of claim 15, further comprising means forchanging the sign of the cross product term if the sign does notcorrespond to the orientation of the at least one polygon.
 17. Theelectronic system of claim 14, wherein the means for evaluatingcomprises means for comparing the sign of the cross product term, asmodified with the orientation decision variable, to determine an actualorientation of the at least one polygon.
 18. The electronic system ofclaim 14, wherein the means for evaluating the sign of the cross productterm and the orientation decision variable comprises means fordetermining whether to cull the data prior to rendering an image of atleast a portion of the digital object.
 19. The electronic system ofclaim 14, wherein the means for sorting and the means for generatingoperate concurrently.
 20. The electronic system of claim 14, furthercomprising means for effecting at least one of a coloring process, ashading process, and a texturing process that employs at least one ofthe cross product term and values of the positional differences betweenthe at least three vertices.
 21. A computer-readable medium havingstored thereon computer-executable instructions, which when executed ona computer, cause the computer to: sort data representative of one ormore polygons comprising at least three vertices the of a digital objectbased on relative positions of the at least three vertices along atleast one axis; calculate a sign of a cross product term of the at leastone polygon after the sorting; generate an orientation decision variablebased on the relative positions of the at least three vertices; andevaluating the sign of the cross product term relative to theorientation decision variable.
 22. The computer-readable medium of claim21, wherein the computer executable instructions executed on a computer,further cause the computer to: determine if each of the one or morepolygons is back facing responsive to the evaluating; cull from thedigital object the back facing polygons; and after the culling,rendering an image of at least a portion of the digital object on adisplay apparatus.